Unable to activate "Messages" on my iPhone

To activate iMessage and FaceTime using your Phone number

1. First, check to know whether you have an SMS plan enabled for at least One SMS*** with your carrier. Contact your Carrier (Cellular Service Provider) the agency that issued the SIM to you
2. If your iPhone is used with a carrier not found in the wireless carrier articles, make sure you can send and receive international SMS. (One international SMS*** Only)
3. If you can't turn on or sign in to iMessage or FaceTime - Apple ...

*** You may not have an SMS plan (Domestic & International) enabled by your carrier for the SIM/eSIM that you use on your device. Apple Inc. does not provide SMS services.

It's possible that you used their Phone Number to send an iMessage, while they might have sent an iMessage using your Apple ID. There's a chance that iMessage isn't activated for the phone number on their device.

iMessage requires phone number activation for sending and receiving messages with other phone numbers. Without activation, you can only chat with people using their Apple IDs.

Activate iMessage and FaceTime using your… - Apple Community